Optimal Markovian coupling for finite activity Lévy processes

We study optimal Markovian couplings of Markov processes, where the optimality is understood in terms of minimization of concave transport costs between evaluations of the coupled processes at corresponding times. We provide explicit constructions of such optimal couplings for one-dimensional finite-activity Lévy processes (continuous-time random walks) whose jump distributions are unimodal but not necessarily symmetric. Remarkably, the optimal Markovian coupling does not depend on the specific concave transport cost. To this end, we combine McCann’s results on optimal transport and Rogers’ results on random walks with a novel uniformization construction that allows us to characterize all Markovian couplings of finite-activity Lévy processes. In particular, we show that the optimal Markovian coupling for finite-activity Lévy processes with non-symmetric unimodal Lévy measures has to allow for non-simultaneous jumps of the two coupled processes.